9 Steps in Maths

Nine Steps or Milestones, A Base or Directions for High School & College Level Mathematics (March 1, 2001)

Some steps may seem too simple for you or too hard. The simpler one's point to difficulties which others have had. The harder ones point to material you should master by yourself or with help. Some steps or sub-steps have to be mastered in sequence, others may be mastered in parallel. See for yourself. Some steps describe material not yet written - plans for tomorrow's student.

Teachers: the steps here point to a redesign of high school or college mathematics which will allow more to go further in an easier fashion.

Step 1: Do all four sets of Arithmetic Review Problems before or at the start of courses in algebra, trigonometry and calculus. These review or reinforce skills for doing arithmetic by hand or with calculators. They may hint at or point to arithmetic or algebraic patterns in computations - situations where two different calculations may give the same result.

Doing arithmetic by hand is important at least for calculations in the absence of an electronic or mechanical calculator. Mastery of arithmetic by hand requires decimal methods to be followed, one step at a time and one step after another, to arrive at results. Here an error in one step makes all the rest suspect or wrong. Knowing that is the first sign of intelligence in following step-by-step methods in any domain, from cooking to science. And except for errors, exact arithmetic with whole numbers and fractions leads to repeatable and reproducible results, independent of the human or mechanical computer.

Marking answers by hand shows the teacher or tutor errors in your notation. Errors in notation indicate miscomprehension. Here the saying, a stitch in time save nines applies. Early correction is best. In general, you should ask someone not to do your homework, but to provide feedback on your errors before you submit your work for correction or marking. Doing this as a matter of habit would lessen the number of errors that you make and lead to a better performance in class. Learn from your errors before you are penalized for them.

Mastery of arithmetic with its methods that lead to repeatable and reproducible is the first source of skill and confidence in mathematics. Awareness that an error in one step makes all the rest suspect is the first sign of careful thought.

Step 2: Read the mathematics-free logic chapters 4, 6, 7, 8 and 12 in Volume 1A. These chapters introduce the Euclidean way of reason with simple words and examples outside of mathematics. Equivalently see chapters 1 to 6 in Volume 2. The Volume 2 version of chapter 4 in 1A is slightly shorter.

The (deductive) Euclidean way of writing and reason employs implication rules, one at a time or one after another, to arrive at conclusions, one at a time or one after another. In these chains of reason, as in arithmetic, an error in one step makes all the rest suspect. And in circumstances, where the implication rules in question apply, the conclusions are independent of the thinker. The chapter on implication rules (4 in volume 1A or 2 in volume 2) is harder than the others. But it explains the difference between one and two-way implication. Not seeing the difference is a source of error in reading or writing or following step by step methods for arriving at conclusions or results. Chapter 6 in 2 (12 in 1A) describes the division of rule-based thought into islands and bodies of "knowledge" in each of which, different starting points may make island points more accessible or not.

The Euclidean way of reason has been previously been introduced within mathematics courses with mathematical examples of chains of reason, examples that may depend on the algebraic shorthand way of writing and reasoning, a further skill that may be hard for many. The treatment here separates the learning and teaching of Euclidean way of writing and writing reasoning from the learning and teaching of the algebraic way of writing and reasoning. The treatment in 4, 6, 7, 8 and 12 in Volume 1A gives an alternative which separates the learning and teaching of Euclidean way of writing and reasoning from the algebraic way of writing and reasoning. So you may learn each way of reason separately before employing both together - divide and conquer.

Mastery of deductive reason with its methods that lead to repeatable and reproducible conclusions could be the second source of skill and confidence in or out of mathematics.

Step 3: Read the essay What is a Variable? and if possible, chapters 7 to 15, in Volume 2, Three Skills for Algebra. The essay and chapters introduce the algebraic way of writing and reasoning with words that have been missing in course design and teacher training programs. The essay and three skills given in the foreword clarify the notion of what is a variable before and besides the uses of notation. The statement that a

letter used in mathematics is a variable, and vice versa,

is half-right and is too misleading, but it is in common use. There are two notions of What is a Variable, one which can mastered before the use of letters and symbols to represent them..

Algebraic and arithmetic expressions are often better read and seen silently than spoken aloud. Words literally have missing for understanding and explaining the shorthand role of notation, symbols and letters, in arithmetic and beyond. But there is a need to talk about numbers, amount and quantities, and about the two shorthand roles of notation. The first role is in describing calculations that might be done. The second role is saying or describing when two different calculations will give the same result. Rules for the latter may be applied one at a time or one after another to solve equations.

The algebraic use of letters and symbols is often declared to be a natural talent, too obvious to explain or impossible to explain directly. In either case, the use is not taught or not understood. This traditional hole in the description of mathematics makes learning and teaching difficult or harder than need-be. Sorry.

Algebraic notation in the first instance is a meta-language for describing arithmetic with expressions and formulas. Algebraic notation in the second instances is a meta-meta-language for saying when different expressions or formulas give the same result.

Mastery of three skills for algebra, two notions what is a variable, and an awareness of the two roles of notation in algebra (describing calculations and rules for saying when calculations are equal) could be a third source of skill and confidence in mathematics. Learning and teaching has been harder than need-be

Step 4: Read (eventually) about [Functions and Sets]

Set membership, union, intersection and complements form a language for a precise description of mathematical ideas. Set theory is emphasized in mathematics since the axiomatic method of analysis, more precisely the arithmetic properties of integers, real numbers and functions can be logically codified and described in it. Outside calculus, you may see the role of set concepts in some presentations of analytic geometry. Lines, planes, surfaces and sometimes solid objects are regarded as set of points. Sets and set membership also have a role in combinatorics and counting, and in the combinatorial based parts of probability. Combinatorics counts or is concerned with the number of ways objects in sets can be grouped or placed together.

The chapter [Functions and Sets] in shows the equivalence of rule- and set-viewpoints of functions. That is never done in Canadian and US course design which stems from the set-theoretic codification or axiomization of mathematics Doing so steps outside the codification.

Step 5: Master the yet to be posted explanation or justification of the decimal methods which you met and hopefully mastered before high school.

The explanation and justification begins the observation the decimal expansions of a whole number is equal to polynomials in powers of ten with coefficients belonging to the set of digits 0 to 9. With simple examples and later with mathematical induction, decimal methods for the addition, subtraction, multiplication and division of whole numbers with carries and borrows etc., may be justified. They may seen as examples of calculations with polynomial with a special conversion or treatment of coefficients when they not in the range 0 to 9.

This topic which links decimal arithmetic to the polynomials is not found in modern course design. But one duty of mathematics education is to provide a thought based understanding of its subject. This topic fills a common gap in course design which leaves explained but not justified decimal methods for arithmetic. Linking decimal arithmetic to the polynomials while showing how the algebraic way of writing and reasoning can be used to explain previously mastered skills, provide motivation and a further context or reason for polynomial operations and the algebraic way of writing and arriving at conclusion about calculations.

Understanding why decimal methods work could be a fourth source of skill and confidence in mathematics. The demonstrations may illustrate polynomial manipulations and beyond that, for enriched students, methods of mathematical induction. (Leave the proofs of associative and distributive/grouping properties needed in these demonstration to a later course in mathematics, one given to students specializing in the subject).

Step 6: The Trigonometry & Complex Number section of the subscriber area describes three ways to cover its subjects. One way, the first, may be enough for you.

The first way described here is for today's students who have mastered trigonometry. It exploits knowledge of trigonometric to arrive at a key property of complex numbers. And trigonometry is not needed if you assume one key property of complex numbers instead of deriving it.

The second and third ways in contrast or reverse the order in which complex numbers and trigonometry. Both ways show how the addition and multiplication of points in the place may geometry extend yours or another's knowledge of arithmetic with unsigned numbers to a full command of real and complex numbers. This command leads to a proof of the Pythagorean theorem and a simpler treatment of trigonometry.. The third way goes beyond the second in providing a knowledge of geometry axioms sufficient to justify the assumption made in the second. The second way with its assumption of key property (see the first way) could be for all students - students in enriched instruction may see the third way

Following a mastery of complex numbers and trigonometry, the trigonometric cosine and sine interpretation of dot- and cross-products follow, and complex number based shortcuts for handling trigonometric identities are easily justified. Engineering students may appreciate the latter.

Putting mastery of complex numbers before trigonometry first provides a quick or enriched way to understand and explain arithmetic with real and complex numbers, and a simpler, quicker, starting part for trigonometry. This could be a fifth source of skill and confidence in mathematics. Learning and teaching has been harder than need-be.

Danger, Danger

The simpler, more effective, but non-traditional, second and third ways to treat complex numbers and trigonometry described may be employed for enriched studies and presented besides traditional approaches. High school teachers in locations where there are central examinations and a common approach to the teaching high school mathematics will not able to present the second and third way to ordinary students. The second and third will provide an easier and better comprehension, but not prepare the student for examination, unless it possible to present the second and third way quickly, and then teach towards the examinations. That is a risk.

Step 7: See an Detailed Explanation of Error Control in Computations: This step for tomorrow's high school student. The book Calculus by Lipman Bers gives a presentation that can be adapted to provide a discussion of significant digits and absolute error control in arithmetic for the operations of addition, subtraction, multiplication and division. This technical discussion of significant digits and absolute error control in calculation is a concrete topic, one that refines or justifies the non-technical discussion or presentation of rules for estimating the number of significant digits, one that optionally can be related to the polynomial perspective of decimal arithmetic in which coefficients outside the range 0 to 9 are handled through carries or a coefficient normalization process. This further discussion of significant digits or accuracy in computations, a version yet to be posted online here, could serve as preparation for the discussion of continuity, limits and convergence in calculus.

Step 8: Explore Why Slopes, A Calculus Preview, and then read Volume 3, Why Slopes and More Math, offline ( online in the subscriber area) to understand further why slopes are met in courses before calculus, or explore the area for a learn or review key ideas in calculus and beyond. Calculus in the first instance is the subject of slope related computations, their reversal and interpretation.

This area rearranges the order of topics in calculus to put the simpler one first and so gradually introducing and reinforcing the skills need for further study, while avoiding calculus or algebra shocks, and providing simple examples to lessen or avoid them. First and further courses in calculus and real analysis switch back and forth between demanding the algebraic way of writing and reasoning. at full strength or not. Before you meet the associated calculus shock, explore Why Slopes, A Calculus Preview

Step 9. Consolidate your knowledge.

Volume 1B, Mathematics Curriculum Notes offers a context for mathematics education, yours or that of others, from elementary school to college in twelve chapters. See the calculus, complex number and algebra appetizers outside the subscriber area of this site first and optionally, most chapters of Volume 1A - some may be too hard.

Consolidation lessons yet to be written has the following tasks.

First describe the algebraic, set theoretic codification of arithmetic in which the existence of sets of real and complex numbers are assumed, along with methods for their arithmetic, and algebraically state assumptions (axioms) which describe properties of arithmetic with real and/or complex numbers. The latter properties should have been assumed or concluded from the earlier geometric discussion of complex numbers and trigonometry.

Second, assume or describe how real number have decimal expansions and assumes or describes the convergence of infinite decimal expansions. The assumptions here handle and sanction the common assumptions about decimals and provides continuity of high mathematics with elementary school mathematics.

Retaining the decimal-free view of the set-theoretic codification of mathematics in courses for students not specializing in the subject obstructs and does not help the common knowledge. Not talking about and not explicitly sanctioning the decimal expansion of real numbers has made learning and teaching harder than need-be, and left a gap in the exposition, a discontinuity between elementary and high school mathematics in North American course designs 1960 to the present.

Third, delicately describes the arithmetic-based codification of geometry in which real numbers may be used as coordinates for a line, and ordered pairs or triplets represents points in a plane or space. That introduces analytic geometry, a way to do or represent geometry without depending on the drawing of suggestive diagrams to arrive at conclusions. This is avoidance of diagrams for proofs, use in illustrations still allowed, provide a thought-based codification of ruler- and compass-based geometry.

The earlier introduction of complex numbers and trigonometry employed suggestive diagrams and assumptions about ruler- and compass-construction. This makes learning and teaching possible. The advanced viewpoint is not for beginners. Yet the advanced viewpoint disowns the earlier, suggestive-drawing, diagram-based introduction to complex numbers, trigonometry and associated parts of calculus, and replaces it with diagram-free, algebraic-arithmetic considerations that most will never see and most students would never be able to follow in the first instance. That is to say that the introduction to complex numbers, trigonometry and calculus requires supports which the advance exposition, a more rigorous approach removes but those same supports also provide a context for the advance exposition.

Fourth, delicately explain that in advance courses, analytic geometry combined with a decimal or decimal-free viewpoint of real numbers can be used to define (to say how to compute) trigonometric functions without the use of diagrams and also how to introduce complex numbers and their arithmetic properties without the use of diagrams. Details of the latter is a story or chain of reason for students in enriched or advanced studies in mathematics, those who take a more rigorous course in real and complex analysis. But the diagram-based explanation is enough for everyone else.

Remark: In geometry based on coordinates or on ruler and compass constructions, the introduction of trigonometric functions using the ratio of sides to triangles or coordinates of points on unit circles represents a large or smaller step away from the development of mathematics from axioms about arithmetic or sets of real numbers. This makes the previous or current development of trigonometry and calculus which use trigonometry, impure. The alternative recommended here gives first and openly, a complete and fully impure geometric development of trigonometry and the properties of real and complex numbers. A switch to the pure axiomatic set-based description of real or complex numbers, or their arithmetic properties comes later.

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